Apparatus and method for estimating maximum road friction coefficient

ABSTRACT

An apparatus and method that determines a maximum road friction coefficient for each wheel regardless of whether the tire is in a predetermined drive slip state, and whether the wheel is a driving wheel. The braking force of each wheel is calculated, and the longitudinal force of the tire of each wheel is calculated. Then, the driving force of the vehicle is calculated, and the lateral force of the tire of each wheel is calculated. Next, the reaction force of the road to each wheel is calculated, and the vertical load of each wheel is calculated. Finally, the ratio of variation in reaction force of the road to variation in composite slip ratio is calculated for each wheel. The sum of the ratio of the reaction force of the road to the vertical load, and the product of a predetermined coefficient and the ratio of variation in reaction force of the road to variation in composite slip ratios is calculated for each wheel as the maximum road friction coefficient.

INCORPORATION BY REFERENCE

[0001] The disclosure of Japanese Patent Application No. 2000-3 53446filed on Nov. 20, 2000 including the specification, drawings andabstract is incorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

[0002] 1. Field of Invention

[0003] The invention generally relates to an estimation of a maximumfriction coefficient between a tire and a road. More particularly, theinvention relates to an apparatus and method for estimating the maximumfriction coefficient between a tire of each wheel and the road, whethereach particular wheel is a driving wheel or a driven wheel.

[0004] 2. Description of Related Art

[0005] An apparatus for estimating the maximum friction coefficientbetween a tire and a road in vehicles such as automobiles is disclosedin, e.g., Japanese Laid-Open Publication No. HEI 3-295445. The apparatusfor estimating the maximum friction coefficient as described in theaforementioned publication calculates a driving torque and a verticalload of a wheel when the wheel is placed into a predeterminedacceleration slip state, and calculates the maximum friction coefficientbetween the road and the tire based on the calculated driving torque andvertical load. This type of maximum friction coefficient estimatingapparatus is known as related art.

[0006] Such a maximum friction coefficient estimating apparatusestimates the maximum friction coefficient based on the driving torqueand the vertical load of the tire at the time the driving wheel isplaced into a predetermined acceleration slip state. Therefore, thisapparatus is capable of accurately estimating the maximum frictioncoefficient between the tire and the road as compared to, for example,an apparatus for estimating a friction coefficient based on the squareroot of the sum of squares of the longitudinal acceleration and thelateral acceleration of the vehicle.

[0007] However, such a conventional maximum friction coefficientestimating apparatus can estimate the maximum friction coefficient onlyat the moment the driving wheel is placed into a predeterminedacceleration slip state. Moreover, in order for this estimatingapparatus to estimate the maximum friction coefficient, the wheel mustbe placed into the predetermined acceleration slip state. Therefore,this estimating apparatus cannot estimate the maximum frictioncoefficient between the tire of the driven wheel and the road.

SUMMARY OF THE INVENTION

[0008] The invention is made in view of the foregoing problems in theconventional maximum friction coefficient estimating apparatus that isconfigured to estimate the maximum friction coefficient based on thedriving torque and the support load of the tire when drive slip occurs.

[0009] As the slip ratio of the tire increases, the road frictioncoefficient gets closer to the maximum friction coefficient, and theratio of variation in reaction force of the road to variation in slipratio gradually gets closer to zero. Moreover, provided that thereaction force of the road to the tire and the vertical load of the tireare obtained, the road friction coefficient (adhesion coefficient) canbe obtained by dividing the reaction force of the road by the supportload. In view of these points, the invention is capable of estimatingthe maximum road friction coefficient regardless of whether the tire isin a predetermined acceleration slip state and whether the wheel is adriving wheel.

[0010] A controller for estimating a maximum friction coefficientaccording to the invention includes: a first section that calculates areaction force of a road to a tire of a wheel based on a model of thetire; a second section that calculates a vertical load of the tire ofthe wheel; a third section that calculates a ratio of the reaction forceof the road to the vertical load as a first ratio; a fourth section thatcalculates as a second ratio a ratio of variation in the reaction forceof the road to variation in a slip ratio of the tire, the slip ratiobeing calculated based on the tire model; and a fifth section thatcalculates a maximum road friction coefficient based on a product of apredetermined coefficient and the second ratio, and the first ratio.

[0011] Thus, the maximum road friction coefficient is calculatedregardless of whether the wheel is in a predetermined acceleration slipstate. Moreover, the maximum road friction coefficient is calculatedeither for the driving wheel or the driven wheel.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012]FIG. 1 is a diagram illustrating the forces applied to each wheelin the longitudinal and lateral directions of the vehicle, and thelongitudinal and lateral forces applied to the vehicle at its center ofgravity;

[0013]FIG. 2 is a diagram illustrating the forces applied to each wheelin the longitudinal and lateral directions of the tire, and thelongitudinal and lateral forces applied to the vehicle at its center ofgravity;

[0014]FIGS. 3A and 3B are graphs showing the relationship between afriction coefficient μ between a road and a tire, and a composite slipratio λ in the case of a common road A and a tire model B of theinvention;

[0015]FIG. 4 is a diagram illustrating a critical friction circle of thetire, moving direction of the tire, and reaction force of the road tothe tire;

[0016]FIG. 5 is a diagram illustrating a method for calculating alateral slip angle β₁ of each wheel based on a lateral slip angle β_(B)and the like of the vehicle;

[0017]FIG. 6 is a diagram illustrating a method for calculating acorrected vehicle speed SVW₁, based on a wheel speed VW₁, of each wheel;

[0018]FIG. 7 is a graph illustrating the gradient (1/F_(Z))(∂F_(XY)/∂λ)in the μ-λ curve;

[0019]FIG. 8 is a graph of the μ-λ curve illustrating a method forcalculating the maximum road friction coefficient μ_(max);

[0020]FIG. 9 is a schematic structural diagram showing a maximumfriction coefficient estimating apparatus applied to a rear-wheel-drivevehicle, according to a first embodiment of the invention;

[0021]FIG. 10 is a flowchart illustrating a routine for estimating themaximum friction coefficient according to the first embodiment;

[0022]FIG. 11 is a flowchart illustrating a subroutine for calculatingthe ratio ∂F_(XY)/∂λ in Step S120 of FIG. 10;

[0023]FIG. 12 is a schematic structural diagram showing a maximumfriction coefficient estimating apparatus applied to a front-wheel-drivevehicle, according to a second embodiment of the invention; and

[0024]FIG. 13 is a flowchart illustrating a routine for estimating themaximum friction coefficient according to the second embodiment.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0025] Before describing embodiments of the invention, an outline of themaximum friction coefficient calculating apparatus according to theinvention will be presented. Note that, for example purposes only, thedescription will be provided for the situation where a vehicle turns tothe left. Regarding the longitudinal force of a tire, driving force isherein regarded as positive force, and braking force is regarded asnegative force. Regarding the longitudinal acceleration, acceleration isregarded as positive acceleration and deceleration is regarded asnegative acceleration. Regarding the lateral force of a tire, leftwardforce is regarded as positive force. Regarding the lateral acceleration,leftward acceleration is regarded as positive acceleration. Regardingthe lateral slip angle of the vehicle, an angle in the counterclockwisedirection is regarded as a positive angle. Regarding the steering angle,an angle in the counterclockwise direction (in the left-handed turningdirection) is regarded as a positive angle.

[0026] 1. Basic Concept

[0027] In FIG. 1, 100_(fl), 100 _(fr), 100 _(rl), and 100 _(rr) denoteleft and right front wheels and left and right rear wheels of a vehicle102, respectively. F_(XV1) (i=fl, fr, rl, rr) denotes the force in thelongitudinal direction of the vehicle that is applied from the road tothe left and right front wheels and left and right rear wheels,respectively. F_(YV1) (i=fl, fr, rl, rr) denotes the force in thelateral direction of the vehicle that is applied from the road to theleft and right front wheels and left and right rear wheels,respectively. F_(XC) and F_(YC) denote the longitudinal force andlateral force that are applied to the vehicle 102 at its center ofgravity 104, respectively.

[0028] As shown in FIG. 1, a balance of the forces in the lateraldirection of the vehicle is given by the following expression (1), andthe vehicle is subjected to the lateral acceleration corresponding tothe lateral force F_(YC):

F _(YC) =F _(YVfl) +F _(YVfr) +F _(YVrl) +F _(YVrr)  (1).

[0029] Similarly, a balance of the forces in the longitudinal directionof the vehicle is given by the following expression (2), and the vehicleis subjected to the longitudinal acceleration corresponding to thelongitudinal force F_(XC):

F _(XC) =F _(XVfl) +F _(XVfr) +F _(XVrl) +F _(XVrr)  (2).

[0030] As shown in FIG. 1, the tread of the vehicle is denoted with Tr,the distance between the front axle and the center of gravity 104 of thevehicle is denoted with Lf, and the distance between the rear axle andthe center of gravity 104 of the vehicle is denoted with Lr. Providedthat the yaw inertia moment of the vehicle is I_(B), and the rate ofchange in yaw rate of the vehicle, i.e., yaw acceleration, is γd,balance of the yaw moment around the center of gravity of the vehicleresulting from the reaction force of the road to the wheels is given bythe following expression (3): $\begin{matrix}{{I_{B}\gamma \quad d} = {{\left( {F_{XVfr} - F_{XVfl} + F_{XVrr} - F_{XVr1}} \right)\frac{Tr}{2}} + {\left( {F_{YVfl} + F_{YVfr}} \right)L_{f}} - {\left( {F_{YVrl} + F_{YVrr}} \right){L_{r}.}}}} & (3)\end{matrix}$

[0031] (1) When I_(B)γd>0:

[0032] The vehicle is subjected to the yaw acceleration in thecounterclockwise direction, so that the lateral slip angle of the rearwheels is increased. The yaw acceleration is gradually reduced in such arange that the sum of the lateral forces on the left and right rearwheels, F_(YVrl)+F_(YVrr), may increase due to the increased lateralslip angle of the rear wheels. The value I_(B)γd finally becomes equalto zero, so that the moment around the center of gravity balancesstatically. Even if the sum of the lateral forces, F_(YVrl)+F_(YVrr),reaches the limit, the moment around the center of gravity cannotbalance statically as long as I_(B)γd>0. Therefore, the vehicle isrendered in a spin state. Note that the limit of the sum(F_(YVrl)+F_(YVrr)) is affected by the longitudinal forces on the leftand right rear wheels F_(YVrl), F_(YVrr), and the maximum road frictioncoefficient of the rear wheels.

[0033] A conventional rear-wheel antilock device and anti wheel-spindevice also have a function to control the braking force and drivingforce of the rear wheels so as to ensure the sum of the lateral forceson the rear wheels, F_(YVrl)+F_(YVrr). However, since these devices arenot intended to optimize the value (F_(YVrl)+F_(YVrr)) such that themoment around the center of gravity balances statically, their spincontrol function is not perfect.

[0034] Therefore, the following operation may be conducted when thevehicle is rendered in a spin state: the braking force and driving forceof each wheel can be controlled so that the moment around the center ofgravity resulting from the reaction force of the road to the wheelsbalances statically. This suppresses and thus eliminates the spin stateof the vehicle.

[0035] (2) When I_(B)γd <0:

[0036] The vehicle is subjected to the yaw acceleration in the clockwisedirection, so that the lateral slip angle of the rear wheels is reduced.The yaw acceleration is therefore gradually reduced. The value I_(B)γdfinally becomes equal to zero, whereby the moment around the center ofgravity balances statically.

[0037] (3) When I_(B)γd=0:

[0038] In this case, the moment around the center of gravity balancesstatically, and the vehicle is in a stable state. Even in such asituation, the turning capability of the vehicle is not effectivelyobtained if the sum of the lateral forces of the left and right frontwheels, F_(YVfl)+F_(YVfr), reaches the limit and the sum of the lateralforces of the left and right rear wheels, F_(YVrl)+F_(YVrr), does notreach the limit. This state is called a drift-out state.

[0039] A conventional front-wheel antilock device controls the brakingforce of the front wheels in order to ensure the sum of the lateralforces on the front wheels, F_(YVfl)+F_(YVrf), As a result, the vehicleis subjected to the yaw acceleration in the counterclockwise direction,whereby the lateral slip angle of the rear wheels is increased. Thefront-wheel antilock device thus increases the sum of the lateral forcesof the rear wheels, F_(YVrl)+F_(YVrr), so as to ensure the turningcapability of the vehicle. However, since this front-wheel antilockdevice is not intended to optimize the value (F_(YVrl)+F_(YVrr)), thedrift-out control function thereof is not perfect.

[0040] Therefore, the following operation may be conducted when thevehicle is rendered in a drift-out state: the braking force and drivingforce of the rear wheels can be controlled so that the lateral slipangle of the rear wheels is increased by the angular moment on thevehicle that is produced by the difference in longitudinal force abetween the left and right rear wheels. The sum of the lateral forces ofthe rear wheels, F_(YVrl)+F_(YVrr), is thus optimized. This improves theturning capability of the vehicle, suppressing and thus eliminating thedrift-out state of the vehicle.

[0041] In order to determine and control the spin state and drift-outstate of the vehicle based on the moment around the center of gravity ofthe vehicle resulting from the reaction force of the road to the wheelsas described above, it is necessary to accurately keep track of thebraking force and driving force of the wheels and the moment resultingfrom the reaction force of the road to the wheels which may cause thespin state and the drift-out state. This requires accurate estimation ofthe maximum road friction coefficient of each wheel for example in themanner described below.

[0042] 2. Basic Process

[0043] It is herein assumed that the longitudinal acceleration of thevehicle is G_(X,) the lateral acceleration of the vehicle is G_(Y), theyaw rate of the vehicle is γ, the yaw acceleration is γd, the steeringangle is δ, the wheel speed of the left and right front wheels and theleft and right rear wheels is VW₁ (i=fl, fr, rl, rr), the wheelacceleration of the left and right front wheels and the left and rightrear wheels is VWd₁ (i=fi, fr, rl, rr), the wheel cylinder hydraulicpressure on the left and right front wheels and the left and right rearwheels is P₁ (i=fl, fr, rl, rr), the lateral slip angle of the vehicleis β_(B) (which is separately calculated as described below), thebraking force of the left and right front wheels and the left and rightrear wheels is B₁ (i=fl, fr, rl, rr), and the vertical load of the leftand right front wheels and the left and right rear wheels is F_(Z1)(i=fl, fr, rl, rr).

[0044] Provided that K_(pf) and K_(pr) respectively represent conversioncoefficients (negative value) from the wheel cylinder hydraulic pressureon the front and rear wheels into the braking force, the braking forcesB_(fl), B_(fr) of the left and right front wheels and the braking forcesB_(rl), B_(rr) of the left and right rear wheels are respectively givenby the following equations (4) to (7):

B _(fl) =K _(Pf) ·P _(fl)  (4)

B _(fr) =K _(pf) ·P _(fr)  (5)

B _(rl) =K _(pr) ·P _(rl)  (6)

B _(rr) =K _(pr) ·P _(rr)  (7).

[0045] Provided that the wheelbase of the vehicle is L (=L_(f)+L_(r)),the height of the center of gravity of the vehicle is h, the vehicleweight is F_(ZV), the gravitational acceleration is g, the roll rigiditydistribution for the front wheels is η_(f), and the roll rigiditydistribution for the rear wheels is η_(r), the vertical loads of theleft and right front wheels and the left and right rear wheels, F_(Zfl),F_(Zfr), F_(Zrl), F_(Zrr), are respectively given by the followingequations (8) to (11): $\begin{matrix}{F_{Zr1} = {\left( {\frac{L_{r} - {hGx}}{2L} - {\eta_{f}\frac{h}{d}G_{Y}}} \right)\quad \frac{F_{ZV}}{g}}} & (8) \\{F_{Zfr} = {\left( {\frac{L_{r} - {hGx}}{2L} + {\eta_{f}\frac{h}{d}G_{Y}}} \right)\quad \frac{F_{ZV}}{g}}} & (9) \\{F_{Zr1} = {\left( {\frac{L_{f} + {hGx}}{2L} - {\eta_{r}\frac{h}{d}G_{Y}}} \right)\quad \frac{F_{ZV}}{g}}} & (10) \\{F_{Zrr} = {\left( {\frac{L_{f} + {hGx}}{2L} - {\eta_{r}\frac{h}{d}G_{Y}}} \right)\quad {\frac{F_{ZV}}{g}.}}} & (11)\end{matrix}$

[0046] 3. Calculation of the Longitudinal Force of the Tire of EachWheel and the Driving Force of the Vehicle

[0047] As shown in FIG. 2, provided that the longitudinal forces oftires of the left and right front wheels and the left and right rearwheels are F_(Xfl), F_(Xfr), F_(Xrl), F_(Xrr), and the lateral forces ofthe tires of the left and right front wheels are F_(Yfl), F_(Yfr), themass of the vehicle is m, and the steering angle is δ, the followingequation (12) is obtained from the balancing of the forces in thelongitudinal direction of the vehicle:

mG _(X)=(F _(Xfl) +F _(Xfl))cosδ−(F _(Yfl) +F _(Yfr))sin δ+(F _(Xrl) +F_(Xrr))  (12).

[0048] (1) A Rear-wheel-drive Vehicle:

[0049] Provided that the effective radius of the tire is r, the drivingforce of the vehicle is D, and the moment of inertia of the left andright front wheels and the left and right rear wheels is I_(W1) (i=fl,fr, rl, rr), the longitudinal forces of the tires of the left and rightfront wheels and the left and right rear wheels, F_(Xfl), F_(Xfr),F_(Xfl), F_(Xrr), are respectively given by the following equations (13)to (16). Note that the wheel acceleration VWd₁ in the equations (13) to(16) and the like may be a derivative value of the corresponding wheelspeed VW₁. $\begin{matrix}{F_{Xf1} = {B_{fl} - \frac{I_{Wf} \cdot {VWd}_{fl}}{r^{2}}}} & (13) \\{F_{Xfr} = {B_{fr} - \frac{I_{Wf} \cdot {VWd}_{fr}}{r^{2}}}} & (14) \\{F_{Xrl} = {B_{rl} + {\frac{1}{2}D} - \frac{I_{Wr} \cdot {VWd}_{r1}}{r^{2}}}} & (15) \\{F_{Xrr} = {B_{rr} + {\frac{1}{2}D} - \frac{I_{Wr} \cdot {VWd}_{rr}}{r^{2}}}} & (16)\end{matrix}$

[0050] From the above equations (12) to (16), the longitudinal forces ofthe tires of the left and right front wheels and the left and right rearwheels, F_(Xfl), F_(Xfr), F_(frl,) F_(Xrr), are respectively given bythe following equations (17) to (20): $\begin{matrix}{F_{Xfl} = {B_{fl} - \frac{I_{Wf} \cdot {VWd}_{fl}}{r^{2}}}} & (17) \\{F_{Xfr} = {B_{fr} - \frac{I_{Wf} \cdot {VWd}_{fr}}{r^{2}}}} & (18) \\{F_{Xrl} = {\frac{1}{2}\left\lbrack {{m\quad G_{x}} - {\left\{ {B_{fl} + B_{fr} - \frac{I_{Wf}\left( {{VWd}_{fl} + {VWd}_{fr}} \right)}{r^{2}}} \right\} \cos \quad \delta} + {\left( {F_{Yfl} + F_{Yfr}} \right)\sin \quad \delta}\quad + \left( {B_{rl} - B_{rr}} \right) - \frac{I_{Wr}\left( {{VWd}_{rl} - {VWd}_{rr}} \right)}{r^{2}}} \right\rbrack}} & (19) \\{F_{Xrr} = {\frac{1}{2}\left\lbrack {{m\quad G_{x}} - {\left\{ {B_{fl} + B_{fr} - \frac{I_{{Wf}{({{VWd}_{fl} + {VWd}_{fr}})}}}{r^{2}}} \right\} \cos \quad \delta} + {\left( {F_{Yfl} + Y_{Yfr}} \right)\sin \quad \delta} + \left( {B_{rr} - B_{rl}} \right) - \frac{I_{Wr}\left( {{VWd}_{rr} - {VWd}_{rl}} \right.}{r^{2}}} \right\rbrack}} & (20)\end{matrix}$

[0051] By substituting the above equations (13) to (16) for the equation(12), the driving force D of the vehicle is obtained by equation (21) asfollows: $\begin{matrix}{D = {{m\quad G_{X}} - {\left\{ {B_{fl} + B_{fr} - \frac{I_{Wf}\left( {{VWd}_{fl} + {VWd}_{fr}} \right)}{r^{2}}} \right\} \cos \quad \delta} + {\left( {F_{Yf1} + F_{Yfr}} \right)\sin \quad \delta} - \left\{ {B_{rl} + B_{rr} - \frac{I_{Wr}\left( {{VWd}_{rl} + {VWd}_{rr}} \right)}{r^{2}}} \right\}}} & (21)\end{matrix}$

[0052] (2) A front-wheel-drive Vehicle:

[0053] In the case of the front-wheel-drive vehicle, the longitudinalforces of the tires of the left and right front wheels and the left andright rear wheels, F_(Xfl), F_(Xfr), F_(Xrl,) F_(Xrr), are respectivelygiven by the following equations (22) to (25): $\begin{matrix}{F_{Xfl} = {B_{fl} + {\frac{1}{2}D} - \frac{I_{Wf} \cdot {VWd}_{fl}}{r^{2}}}} & (22) \\{F_{Xfr} = {B_{fr} + {\frac{1}{2}D} - \frac{I_{Wf} \cdot {VWd}_{fr}}{r^{2}}}} & (23) \\{F_{Xrl} = {B_{rl} - \frac{I_{Wr} \cdot {VWd}_{rl}}{r^{2}}}} & (24) \\{F_{Xrr} = {B_{rr} - {\frac{I_{Wr} \cdot {VWd}_{rr}}{r^{2}}.}}} & (25)\end{matrix}$

[0054] From the above equations (12) and (22) to (25), the longitudinalforces of the tires of the left and right front wheels and the left andright rear wheels, F_(Xfl), F_(Xfr), F_(Xrl), F_(Xrr), are respectivelygiven by the following equations (26) to (29): $\begin{matrix}{F_{Xfl} = {\frac{{m\quad G_{x}} + {\left( {F_{Yfl} + F_{Yfr}} \right)\sin \quad \delta} - \left( {B_{rl} + B_{rr}} \right) + \frac{I_{Wr}\left( {{VWd}_{rl} + {VWd}_{rr}} \right)}{r^{2}}}{2\quad \cos \quad \delta} + {\frac{1}{2}\left( {B_{fl} - B_{fr}} \right)} - \frac{I_{Wr}\left( {{VWd}_{fl} - {VWd}_{fr}} \right)}{2r^{2}}}} & (26) \\{F_{Xfr} = {\frac{{m\quad G_{x}} + {\left( {F_{Yfl} + F_{Yfr}} \right)\sin \quad \delta} - \left( {B_{rl} + B_{rr}} \right) + \frac{I_{Wr}\left( {{VWd}_{rl} + {VWd}_{rr}} \right)}{r^{2}}}{2\quad \cos \quad \delta} - {\frac{1}{2}\left( {B_{fl} - B_{fr}} \right)} + \frac{I_{Wr}\left( {{VWd}_{fl} - {VWd}_{fr}} \right)}{2r^{2}}}} & (27) \\{F_{Xrl} = {B_{rl} - \frac{I_{Wr} \cdot {VWd}_{rl}}{r^{2}}}} & (28) \\{F_{Xrr} = {B_{rr} - {\frac{I_{Wr} \cdot {VWd}_{rr}}{r^{2}}.}}} & (29)\end{matrix}$

[0055] By substituting the above equations (22) to (25) for the equation(12), the driving force D of the vehicle can be obtained by equation(30) as follows: $\begin{matrix}{D = {\frac{{m\quad G_{x}} + {\left( {F_{Yfl} + F_{Yfr}} \right)\sin \quad \delta} - \left( {B_{rl} + B_{rr}} \right) + \frac{I_{Wr}\left( {{VWd}_{rl} + {VWd}_{rr}} \right)}{r^{2}}}{\cos \quad \delta} - \left( {B_{fl} + B_{fr}} \right) + {\frac{I_{Wr}\left( {{VWd}_{fl} + {VWd}_{fr}} \right)}{r^{2}}.}}} & (30)\end{matrix}$

[0056] As can be seen from the foregoing description, by using theprevious calculated values of the lateral forces of the tires of thefront wheels F_(Yfl) and F_(Yfr) in the above equations, thelongitudinal acceleration of the vehicle G_(X), the steering angle δ,the brake hydraulic pressure P₁ of each wheel, and the wheelacceleration VWd₁, are detected. Accordingly, the longitudinal force ofthe tire of each wheel, F_(X1), is calculated according to the equations(17) to (20) or the equations (26) to (29). In this case, the engine andthe driving system need not be taken into account even when driving thevehicle. Moreover, the driving force that is transmitted from the engineto the axle of the driving wheels through the driving system can becalculated according to the above equation (21) or (30). In this case,the driving force of the axle of the driving wheels can be calculatedwithout taking into account the engine map, gear ratio of the drivingsystem, and transmission efficiency.

[0057] 4. Calculation of the Lateral Force of the Tire of Each Wheel

[0058] The following equations (31) and (32) are obtained from thebalancing of the forces in the lateral direction of the vehicle and thebalancing of the yaw moment around the center of gravity:

mG _(Y) =F _(YVfl) +F _(YVfr) +F _(YVrl) +F _(YVrr)  (31)

[0059] $\begin{matrix}{{I_{B}\gamma \quad d} = {{\frac{Tr}{2}\left( {F_{XVfr} - F_{XVfl}} \right)} + {L_{f}\left( {F_{YVfl} + F_{YVfr}} \right)} + {\frac{Tr}{2}\left( {F_{XVrr} - F_{XVrl}} \right)} - {{L_{r}\left( {F_{YVrl} + F_{YVrr}} \right)}.}}} & (32)\end{matrix}$

[0060] (1) The Lateral Forces of the Tires of the Front Wheels:

[0061] The above equation (32) is rewritten as equation (33) as follows:$\begin{matrix}{\begin{matrix}{{I_{B}\gamma \quad d} = \quad {{\frac{Tr}{2}\left( {F_{XVfr} - F_{XVfl}} \right)} + {L_{f}\left( {F_{YVfl} + F_{YVfr}} \right)} +}} \\{\quad {{\frac{Tr}{2}\left( {F_{XVrr} - F_{XVrl}} \right)} - {L_{r}\left( {{m\quad G_{Y}} - F_{YVfl} - F_{YVfr}} \right)}}} \\{= \quad {{\frac{Tr}{2}\left( {F_{XVfr} - F_{XVfl} + F_{XVrr} - F_{XVrl}} \right)} +}} \\{\quad {{L\left( {F_{YVfl} + F_{YVfr}} \right)} - {L_{r}m\quad G_{Y}}}}\end{matrix}.} & (33)\end{matrix}$

[0062] Substituting the following equations (34) to (37) for theequation (33) results in the following equation (38):

F _(XVfl) =F _(Xfl) cosδ−F _(Yfl) sinδ  (34)

F _(XVfr) =F _(Xfr) cosδ−F _(Yfr) sinδ  (35)

F _(YVfl) =F _(Xfl) sinδ+F _(Yfl) cosδ  (36)

F _(YVfr) =F _(Xfr) sinδ+F _(Xfr) cosδ  (37)

[0063] $\begin{matrix}{{{\left( {{\cos \quad \delta} + {\frac{Tr}{2L}\sin \quad \delta}} \right)F_{Yfl}} + {\left( {{\cos \quad \delta} - {\frac{Tr}{2L}\sin \quad \delta}} \right)F_{Yfr}}} = {\frac{{I_{B}\gamma \quad d} + {L_{r}m\quad G_{Y}} - {\frac{Tr}{2}\left( {F_{Xrr} - F_{Xrl}} \right)}}{L} - {\left( {{\sin \quad \delta} - {\frac{Tr}{2L}\cos \quad \delta}} \right)F_{Xfl}} - {\left( {{\sin \quad \delta} + {\frac{Tr}{2L}\cos \quad \delta}} \right){F_{Xfr}.}}}} & (38)\end{matrix}$

[0064] Provided that the respective coefficients of the lateral forceson tires F_(Yfl, F) _(Yfr) in the equation (38) are A_(k) and B_(k) andthe right side of the equation (38) is C_(k), the equation (38) isrewritten as the following equation (39). Note that, in the practicalrange of the steering angle, A_(k)>0 and B_(k)>0.

A _(k) ·F _(Yfl) +B _(k) ·F _(Yfr) =C _(k)  (39)

[0065] In general, the ratio of the reaction force of the road betweenthe left and right front wheels corresponds to the ratio of the verticalloads between the left and right front wheels (or the ratio between theproducts of the maximum road friction coefficient and the respectivevertical loads). Therefore, the following equation (40) is obtained:$\begin{matrix}{{\left( {F_{Xfl}^{2} + Y_{Yfl}^{2}} \right)\quad \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}} = {F_{Xfr}^{2\quad} + {F_{Yfr}^{2}.}}} & (40)\end{matrix}$

[0066] Substituting Fyfr in the equation (39) for the equation (40)results in the following equation (41), whereby the following equation(42) is obtained: $\begin{matrix}{{\left( {F_{Xfl}^{2} + Y_{YFl}^{2}} \right)\quad \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}} = {{F_{Xfr}^{2} + {\left( \frac{C_{k} - {A_{k} \cdot F_{Yfl}}}{B_{k}} \right)^{2}\left\{ {\left( \frac{A_{k}}{B_{k}} \right)^{2} - \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}} \right\} F_{Yfl}^{2}} - {\frac{2A_{k}C_{k}}{B_{k}^{2}}F_{Yfl}} + \left( \frac{C_{k}}{B_{k}} \right)^{2} + F_{Xfr}^{2} - {\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}F_{Xfl}^{2}}} = 0}} & (41) \\{F_{Yfl} = {\frac{\frac{A_{k}C_{k}}{B_{k}^{2}} \pm \sqrt{\begin{matrix}{{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}\left\{ {\left( \frac{C_{k}}{B_{k}} \right)^{2} + F_{Xfr}^{2} + {\left( \frac{A_{k}}{B_{k}} \right)^{2}F_{Xfl}^{2}}} \right\}} -} \\{{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{4}F_{Xfl}^{2}} - {\left( \frac{A_{k}}{B_{k}} \right)^{2}F_{Xfr}^{2}}}\end{matrix}}}{\left( \frac{A_{k}}{B_{k}} \right)^{2} - \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}}.}} & (42)\end{matrix}$

[0067] Similar, substituting F_(Yfl) in the equation (39) for theequation (40) results in the following equation (43): $\begin{matrix}{F_{Yfr} = {\frac{\frac{B_{k}C_{k}}{A_{k}^{2}} \pm \sqrt{\begin{matrix}{{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}\left\{ {\left( \frac{C_{k}}{A_{k}} \right)^{2} + F_{Xfl}^{2} + {\left( \frac{B_{k}}{A_{k}} \right)^{2}F_{Xfr}^{2}}} \right\}} -} \\{{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{4}F_{Xfr}^{2}} - {\left( \frac{B_{k}}{A_{k}} \right)^{2}F_{Xfl}^{2}}}\end{matrix}}}{\left( \frac{B_{k}}{A_{k}} \right)^{2} - \left( \frac{F_{Zfl}}{F_{Zfr}} \right)^{2}}.}} & (43)\end{matrix}$

[0068] While the vehicle is turning to the left, C_(k)>0, F_(Yfl)>0 andF_(Yfr)>0. When the following expression (44) is satisfied, thedenominator in the above equation (43) is negative. In order to satisfyF_(Yrr)>0, the sign “±” in the equation (43) must be negative “−”.Accordingly, the lateral force of the tire of the right front wheel,F_(Yfr,) is obtained by the following equation (45), and the lateralforce of the tire of the left front wheel, F_(Yfl), is obtained by thefollowing equation (46): $\begin{matrix}{{\left( \frac{A_{k}}{B_{k}} \right)^{2} - \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}} > 0} & (44) \\{F_{Yfr} = \frac{\frac{B_{k}C_{k}}{A_{k}^{2}} - \sqrt{\begin{matrix}{\left( \frac{F_{Zfl}}{F_{Zfr}} \right)^{2}\left\{ {\left( \frac{C_{k}}{A_{k}} \right)^{2} + F_{Xfl}^{2} +} \right.} \\{\left. {\left( \frac{B_{k}}{A_{k}} \right)^{2}F_{Xfr}^{2}} \right\} - {\left( \frac{F_{Zfl}}{F_{Zfr}} \right)^{4}F_{Xfr}^{2}} -} \\{\left( \frac{B_{k}}{A_{k}} \right)^{2}F_{Xfl}^{2}}\end{matrix}}}{\left( \frac{B_{k}}{A_{k}} \right)^{2} - \left( \frac{F_{Zfl}}{F_{Zfr}} \right)^{2}}} & (45) \\{F_{Yfl} = {\frac{C_{k} - {B_{k} \cdot F_{Yfr}}}{A_{k}}.}} & (46)\end{matrix}$

[0069] When the following expression (47) is satisfied, the denominatorin the above equation (42) is negative. Therefore, in order to satisfyF_(Yrl)>0, the sign “±” in the equation (42) must be negative “−”.Accordingly, the lateral force of the tire of the left front wheel,F_(Yfl), is obtained by the following equation (48), and the lateralforce of the tire of the right front wheel, F_(Yfr), is obtained by thefollowing equation (49): $\begin{matrix}{{\left( \frac{A_{k}}{B_{k}} \right)^{2} - \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}} < 0} & (47) \\{F_{Yfl} = \frac{\frac{B_{k}C_{k}}{B_{k}^{2}} - \sqrt{\begin{matrix}{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}\left\{ {\left( \frac{C_{k}}{B_{k}} \right)^{2} + F_{Xfr}^{2} +} \right.} \\{\left. {\left( \frac{A_{k}}{B_{k}} \right)^{2}F_{Xfl}^{2}} \right\} - {\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{4}F_{Xfl}^{2}} -} \\{\left( \frac{A_{k}}{B_{k}} \right)^{2}F_{Xfr}^{2}}\end{matrix}}}{\left( \frac{A_{k}}{B_{k}} \right)^{2} - \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}}} & (48) \\{F_{Yfr} = {\frac{C_{k} - {A_{k} \cdot F_{Yfl}}}{B_{k}}.}} & (49)\end{matrix}$

[0070] While the vehicle is turning to the right, C_(k)<0, F_(Yfl)<0 andF_(Yfr)<0. When the above expression (44) is satisfied, the denominatorin the above equation (43) is negative. Therefore, in order to satisfyF_(Yfr)<0, the sign “±” in the equation (43) must be positive “+”.Accordingly, the lateral force of the tire of the right front wheel,F_(Yfr), is obtained by the following equation (50), and the lateralforce of the tire of the left front wheel, F_(Yfl), is obtained by thefollowing equation (51): $\begin{matrix}{F_{Yfr} = \frac{\frac{B_{k}C_{k}}{A_{k}^{2}} - \sqrt{\begin{matrix}{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}\left\{ {\left( \frac{C_{k}}{A_{k}} \right)^{2} + F_{Xfl}^{2} +} \right.} \\{\left. {\left( \frac{A_{k}}{B_{k}} \right)^{2}F_{Xfr}^{2}} \right\} - {\left( \frac{F_{Zfl}}{F_{Zfr}} \right)^{4}F_{Xfr}^{2}} -} \\{\left( \frac{B_{k}}{A_{k}} \right)^{2}F_{Xfl}^{2}}\end{matrix}}}{\left( \frac{B_{k}}{A_{k}} \right)^{2} - \left( \frac{F_{Zfl}}{F_{Zfr}} \right)^{2}}} & (50) \\{F_{Yfl} = {\frac{C_{k} - {B_{k} \cdot F_{Yfr}}}{A_{k}}.}} & (51)\end{matrix}$

[0071] When the above expression (47) is satisfied, the denominator inthe above equation (42) is negative. Therefore, in order to satisfyF_(Yfl)<0, the sign “+” in the equation (42) must be positive “+”.Accordingly, the lateral force of the tire of the left front wheel,F_(Yfl), is obtained by the following equation (52), and the lateralforce of the tire of the right front wheel, F_(Yfr), is obtained by thefollowing equation (53): $\begin{matrix}{F_{Yfl} = \frac{\frac{A_{k}C_{k}}{B_{k}^{2}} - \sqrt{\begin{matrix}{\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}\left\{ {\left( \frac{C_{k}}{B_{k}} \right)^{2} + F_{Xfr}^{2} +} \right.} \\{\left. {\left( \frac{A_{k}}{B_{k}} \right)^{2}F_{Xfl}^{2}} \right\} - {\left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{4}F_{Xfl}^{2}} -} \\{\left( \frac{B_{k}}{A_{k}} \right)^{2}F_{Xfr}^{2}}\end{matrix}}}{\left( \frac{B_{k}}{A_{k}} \right)^{2} - \left( \frac{F_{Zfr}}{F_{Zfl}} \right)^{2}}} & (52) \\{F_{Yfr} = {\frac{C_{k} - {A_{k} \cdot F_{Yfl}}}{B_{k}}.}} & (53)\end{matrix}$

[0072] (2) The lateral Forces of the Tires of the Rear Wheels:

[0073] The above equation (32) is rewritten as equation (54) as follows:$\begin{matrix}{{I_{B}\gamma \quad d} = {{{\frac{Tr}{2}\left( {F_{XVfr} - F_{XVfl}} \right)} + {L_{f}\left( {F_{YVfl} + F_{YVfr}} \right)} + {\frac{Tr}{2}\left( {F_{XVrr} - F_{XVrl}} \right)} - {L_{r}\left( {F_{YVrl} + F_{YVrr}} \right)}} = {{\frac{Tr}{2}\left( {F_{XVfr} - F_{XVfl}} \right)} + {L_{f}\left( {{m\quad G_{Y}} - F_{YVrl} - F_{YVrr}} \right)} + {\frac{Tr}{2}\left( {F_{XVrr} - F_{XVrl}} \right)} - {{L_{r}\left( {F_{YVrl} + F_{YVrr}} \right)}.}}}} & (54)\end{matrix}$

[0074] Substituting the following equations (55) to (60) for theequation (54) results in the following equation (61). Note that thevalues calculated in “(1) The lateral forces of the tires of the frontwheels” are used F_(Yfl) and F_(Yfr) in the equations (55) and (56).

F _(XVfl) =F _(Xfl) cosδ−F _(Yfl) sinδ  (55)

F _(XVfr) =F _(Xfr) cosδ−F _(Yfr) sinδ  (56)

F _(XVrl) =F _(Xrl)  (57)

F _(XVrr) =F _(Xrr)  (58)

F _(YVrl) =F _(Yrl)  (59)

F _(YVrl) =F _(YVrr)  (60)

[0075] $\begin{matrix}{{F_{Yrl} + F_{Yrr}} = \frac{\begin{matrix}{{{- I_{B}}\gamma \quad d} + {L_{f}m\quad G_{Y}} + {\frac{Tr}{2}\left\{ {{\left( {F_{Xfr} - F_{Xfl}} \right)\cos \quad \delta} -} \right.}} \\\left. {\left( {F_{Yfr} - F_{Yfl}} \right)\sin \quad \delta} \right\}\end{matrix}}{L}} & (61)\end{matrix}$

[0076] Provided that the right side of the equation (61) is D_(k), theequation (61) is rewritten as equation (62) as follows:

F _(Yrl) +F _(Yrr) =D _(k)  (62).

[0077] In general, the ratio of the reaction force of the road betweenthe left and right rear wheels also corresponds to the ratio of thevertical loads between the left and right rear wheels (or the ratio ofthe products of the maximum road friction coefficient and the respectivevertical loads). Therefore, the following equations (63) and (64) areobtained: $\begin{matrix}{{\left( {F_{Xrl}^{2} + F_{Yrl}^{2}} \right)\quad \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}} = {{F_{Xrr}^{2} + F_{Yrr}^{2}} = {F_{Xrr}^{2} + \left( {D_{k} - F_{Yrl}} \right)^{2}}}} & (63) \\{{{\left\{ {1 - \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}} \right\} F_{Yrl}^{2}} - {2D_{k}F_{Yrl}} + D_{k}^{2} + F_{Xrr}^{2} - {\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}F_{Xrl}^{2}}} = 0.} & (64)\end{matrix}$

[0078] Substituting F_(Yrr) in the above equation (62) for the equation(64) results in the following equation (65): $\begin{matrix}{F_{Yfl} = {\frac{D_{k} \pm \sqrt{\begin{matrix}{{\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}\left( {D_{k}^{2} + F_{Xrl}^{2} + F_{Xrr}^{2}} \right)} - F_{Xrr}^{2} -} \\{\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{4} \cdot F_{Xrl}^{2}}\end{matrix}}}{1 - \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}}.}} & (65)\end{matrix}$

[0079] Similarly, substituting F_(Yrl) in the above equation (62) forthe equation (64) results in the following equation (66):$\begin{matrix}{F_{Yrr} = {\frac{D_{k} \pm \sqrt{\begin{matrix}{{\left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{2}\left( {D_{k}^{2} + F_{Xrl}^{2} + F_{Xrr}^{2}} \right)} - F_{Xrl}^{2} -} \\{\left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{4} \cdot F_{Xrr}^{2}}\end{matrix}}}{1 - \left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{2}}.}} & (66)\end{matrix}$

[0080] While the vehicle is turning to the left, D_(k)>0,F_(Yrl)>0 andF_(Yrr)>0. When the following expression (67) is satisfied, thedenominator in the above equation (66) is negative. Therefore, in orderto satisfy F_(Yrr)>0, the sign “±”, in the equation (66) must benegative “−”. Accordingly, the lateral force of the tire of the rightrear wheel, F_(Yrr) is obtained by the following equation (68), and thelateral force of the tire of the left rear wheel, F_(Yrl), is obtainedby the following equation (69): $\begin{matrix}{{1 - \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}} > 0} & (67) \\{F_{Yrr} = \frac{D_{k} - \sqrt{\begin{matrix}{{\left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{2}\left( {D_{k}^{2} + F_{Xrl}^{2} + F_{Xrr}^{2}} \right)} - F_{Xrl}^{2} -} \\{\left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{4}F_{Xrr}^{2}}\end{matrix}}}{1 - \left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{2}}} & (68)\end{matrix}$

F _(Yrl) =D _(k) −F _(Yrr)  (69).

[0081] When the following expression (70) is satisfied, the denominatorin the above equation (65) is negative. Therefore, in order to satisfyF_(Yrl)>0, the sign “±” in the equation (65) must be negative “−”.Accordingly, the lateral force of the tire of the left rear wheel,F_(Yrl), is obtained by the following equation (71), and the lateralforce of the tire of the right rear wheel, F_(Yrr) is obtained by thefollowing equation (72): $\begin{matrix}{{1 - \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}} < 0} & (70) \\{F_{Yrl} = \frac{D_{k} - \sqrt{\begin{matrix}{{\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}\left( {D_{k}^{2} + F_{Xrl}^{2} + F_{Xrr}^{2}} \right)} - F_{Xrr}^{2} -} \\{\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{4}F_{Xrl}^{2}}\end{matrix}}}{1 - \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}}} & (71)\end{matrix}$

F _(Yrr) =D _(k) −F _(Yrl)  (72).

[0082] While the vehicle is turning to the right, D_(k)<0, F_(Yrl)<0 andF_(Yrl)<0. When the above expression (67) is satisfied, the denominatorin the above equation (66) is negative. Therefore, in order to satisfyF_(Yrr)<0, the sign “+” in the equation (66) must be positive “+”.Accordingly, the lateral force of the tire of the right rear wheel,F_(YVrr), is obtained by the following equation (73), and the lateralforce of the tire of the left rear wheel, F_(YVrl), is obtained by thefollowing equation (74): $\begin{matrix}{F_{Yrr} = \frac{D_{k} + \sqrt{\begin{matrix}{{\left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{2}\left( {D_{k}^{2} + F_{Xrl}^{2} + F_{Xrr}^{2}} \right)} - F_{Xrl}^{2} -} \\{\left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{4}F}\end{matrix}}}{1 - \left( \frac{F_{Zrl}}{F_{Zrr}} \right)^{2}}} & (73)\end{matrix}$

F _(Yrl) =D _(k) −F _(Yrr)  (74).

[0083] When the above expression (70) is satisfied, the denominator inthe above equation (65) is negative. Therefore, in order to satisfyF_(Yrl)<0, the sign “±” in the equation (65) must be positive “+”.Accordingly, the lateral force of the tire of the left rear wheel,F_(Yrl), is obtained by the following equation (75), and the lateralforce of the tire of the right rear wheel, F_(YVrr), is obtained by thefollowing equation (76): $\begin{matrix}{F_{Yrl} = \frac{D_{k} + \sqrt{\begin{matrix}{{\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}\left( {D_{k}^{2} + F_{Xrl}^{2} + F_{Xrr}^{2}} \right)} - F_{Xrr}^{2} -} \\{\left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{4}F_{Xrl}^{2}}\end{matrix}}}{1 - \left( \frac{F_{Zrr}}{F_{Zrl}} \right)^{2}}} & (75)\end{matrix}$

F _(Yrr) =D _(k) −F _(Yrl)  (76).

[0084] 5. Calculation of the Reaction Force of the Road to the Tires

[0085] The reaction force of the road to the tire of each wheel,F_(XY1), (i=fl, fr, rl, rr) is obtained as the resultant force of thelongitudinal force F_(X1) and the lateral force F_(Y1), (i.e., theresultant reaction force of the road) by the following equations (77) to(80):

F _(XYfl) ={square root}{square root over (F_(Xfl) ²+F_(Yfl) ²)}  (77)

F _(XYfr) ={square root}{square root over (F_(Xfr) ²+F_(Yfr) ²)}  (78)

F _(XYrl) ={square root}{square root over (F_(Xrl) ²+F_(Yrl) ²)}  (79)

F _(XYrr) ={square root}{square root over (F_(Xrr) ²+F_(Yrr) ²)}  (80).

[0086] 6. Tire Model (Part 1)

[0087] According to the “brush tire model” (the equations upon drivingin the section 2 above) described in “Vehicle Dynamics and Control”(Masato ABE, Sankaido), provided that VB is a vehicle speed, β is alateral slip angle of the tire, K_(β)is lateral rigidity of the tire,K_(S) is longitudinal rigidity of the tire, μ_(max) is the maximum roadfriction coefficient and F_(Z) is vertical load of the tire, a slipratio S and a composite slip ratio λ are respectively given by thefollowing equations (81) and (82). Moreover, ξ is defined by thefollowing equation (83): $\begin{matrix}{S = \frac{{VB} - {VW}}{VW}} & (81) \\{\lambda = \sqrt{S^{2} + {\left( {1 + S} \right)^{2}\left( \frac{K_{\beta}}{K_{S}} \right)^{2}\tan^{2}\beta}}} & (82) \\{\xi = {1 - {\frac{K_{S}}{3\mu_{\max}F_{Z}}\lambda}}} & (83)\end{matrix}$

[0088] Note that the composite slip ratio λ is a slip ratio in thedirection along the reaction force F_(XY1) of the road to the tire. Ingeneral, the relation between the friction coefficient μ between thetire and the road and the composite slip ratio λ is as shown in FIG. 3A.According to the tire model, however, the relation between the frictioncoefficient μ and the composite slip ratio λ is as shown in FIG. 3B, andthe maximum road friction coefficient μ_(max) is defined as shown inFIG. 3B.

[0089] When ξ>0, the longitudinal force F_(X) and the lateral force F,of the tire are respectively given by the following equations (84) and(85), provided that the reaction force of the road to the tire isapplied at an angle θ with respect to the longitudinal direction of thetire: $\begin{matrix}{F_{X} = {{{- K_{S}}S\quad \xi^{2}} - {6\mu_{\max}F_{Z}\cos \quad \theta \quad \left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}}} & (84) \\{F_{Y} = {{{- {K_{\beta}\left( {1 + S} \right)}}\quad \tan \quad {\beta \cdot \xi^{2}}} - {6\mu_{\max}F_{Z}\sin \quad {{\theta \left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}.}}}} & (85)\end{matrix}$

[0090] When ξ<0, the longitudinal force F_(X) and the lateral force Fyof the tire are respectively given by the following equations (86) and(87), where cosθand sinθ are respectively given by the followingequations (88) and (89):

F _(X)=−μ_(max) F _(Z) cosθ  (86)

F _(Y)=−μ_(max) F _(Z) sinθ  (87)

[0091] $\begin{matrix}{{\cos \quad \theta} = \frac{S}{\lambda}} & (88) \\{{\sin \quad \theta} = \frac{K_{\beta}\tan \quad {\beta \cdot \left( {1 + S} \right)}}{K_{S}\lambda}} & (89)\end{matrix}$

[0092] The foregoing description is given by the aforementionedpublication. The above equations (84) and (85) can be respectivelyrewritten as the following equations (90) and (91): $\begin{matrix}{F_{X} = {{{{- K_{S}}S\quad \xi^{2}} - {6\quad \mu_{\max}\quad F_{Z}\quad \cos \quad {\theta \cdot \left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}}} = {{- K_{S}}S\quad \left\{ {\xi^{2} + {\frac{6\quad \mu_{\max}\quad F_{Z}}{K_{S}\quad \lambda}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}}}} & (90) \\{F_{Y} = {{{{- K_{\beta}}\quad \left( {1 + S} \right)\quad \tan \quad {\beta \cdot \xi^{2}}} - {6\quad \mu_{\max}\quad F_{Z}\sin \quad {\theta \cdot \left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}}} = {{- K_{\beta}}\quad \left( {1 + S} \right)\quad \tan \quad \beta {\left\{ {\xi^{2} + {\frac{6\quad \mu_{\max}\quad F_{Z}}{K_{S}\quad \lambda}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}.}}}} & (91)\end{matrix}$

[0093] Accordingly, the reaction force of the road to the tire, F_(XY),can be given by the following equation (93) based on the followingequation of squares (92): $\begin{matrix}{F_{XY}^{2} = {{F_{X}^{2} + F_{Y}^{2}} = {{\left\{ {{K_{S}^{2}S^{2}} + {K_{\beta}^{2}\quad \left( {1 + S} \right)^{2}\tan^{2}\beta}} \right\} \quad \left\{ {\xi^{2} + {\frac{6\quad \mu_{\max}\quad F_{Z}}{K_{S}\quad \lambda}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}^{2}} = {K_{S}^{2}\quad \lambda^{2}\left\{ {\xi^{2} + {\frac{6\quad \mu_{\max}\quad F_{Z}}{K_{S}\quad \lambda}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}^{2}}}}} & (92) \\{F_{XY} = {K_{S}\quad \lambda {\left\{ {\xi^{2} + {\frac{6\quad \mu_{\max}\quad F_{Z}}{K_{S}\quad \lambda}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}.}}} & (93)\end{matrix}$

[0094] The equation (93) and the equations (90), (91) lead to thefollowing equations (94) and (95). The longitudinal force F_(X) and thelateral force F_(Y) of the tire can thus be obtained from theseequations: $\begin{matrix}{F_{X} = {{{- \frac{K_{S}\quad S}{K_{S}\quad \lambda}}\quad F_{XY}} = {{- \frac{S}{\lambda}}\quad F_{XY}}}} & (94) \\{F_{Y} = {{{- \frac{K_{\beta}\quad \left( {1 + S} \right)\quad \tan \quad \beta}{K_{S}\quad \lambda}}\quad F_{XY}} = {{- \frac{K_{\beta}}{K_{S}}}\quad \tan \quad {\beta \cdot \frac{1 + S}{\lambda}}{F_{XY}.}}}} & (95)\end{matrix}$

[0095] From the above equation (83), the composite slip ratio λ is givenby the following equation (96). By substituting the composite slip ratioλ for the above equation (93), the reaction force of the road to thetire, F_(XY), is obtained as the following equation (97):$\begin{matrix}{\lambda = {\left( {1 - \xi} \right)\quad \frac{3\quad \mu_{\max}F_{Z}}{K_{S}}}} & (96) \\{F_{XY} = {{\left( {1 - \xi} \right)\quad 3\quad \mu_{\max}F_{Z}\left\{ {\xi^{2} + {\frac{6\quad \mu_{\max}\quad F_{Z}}{{K_{S}\quad \left( {1 - \xi} \right)\quad \frac{3\quad \mu_{\max}F_{Z}}{K_{S}}}\quad}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}} = {{\left( {1 - \xi} \right)\quad 3\quad \mu_{\max}F_{Z}\quad \left\{ {\xi^{2} + {\frac{2}{\left( {1 - \xi} \right)}\left( {\frac{1}{6} - {\frac{1}{2}\xi^{2}} + {\frac{1}{3}\xi^{3}}} \right)}} \right\}} = {\mu_{\max}{F_{Z}\left( {1 - \xi^{3}} \right)}}}}} & (97)\end{matrix}$

[0096] The following equation (98) is obtained from the above equations(94) and (95), and the following equation (99) is obtained from theabove equation (97): $\begin{matrix}{F_{Y} = {{\frac{1 + S}{S}\quad \frac{K_{\beta}}{K_{S}}\quad \tan \quad {\beta \cdot F_{X}}\quad \frac{\partial F_{XY}}{\partial\lambda}} = {{\frac{\partial F_{XY}}{\partial\xi}\frac{\partial\xi}{\partial\lambda}} = {{- 3}\quad \mu_{\max}F_{Z}\quad {\xi^{2} \cdot {- \frac{K_{S}}{3\quad \mu_{\max}F_{Z}}}}}}}} & (98)\end{matrix}$

=K _(S)ξ²  (99).

[0097]FIG. 4 shows a critical friction circle 108 of a tire 106. Arrow110 indicates the moving direction of the tire. The points A and Crespectively indicate intersections of the critical friction circle 108and lines 114, 116. The line 114 extends in the longitudinal directionof the tire, the line 116 extends in the lateral direction of the tire,and both lines 114, 116 pass through a ground contact point 112 of thetire. The point E indicates an intersection of the moving direction 110of the tire and the critical friction circle 108. The points B and Drespectively indicate the points on a perfect circle 118, which arelocated closest to point C among the points on the critical frictioncircle 108.

[0098] When ξ>0, the above equations (94), (95), (97), (98) and (99)respectively represent the values in the case where the tip of thevector of the reaction force of the road to the tire, F_(XY),is locatedbetween the points B and D on the critical friction circle 108.

[0099] When ξ<0, the reaction force of the road to the tire, F_(XY), isgiven by the following equation (100) based on the above equations (84)and (85). The longitudinal force of the tire, F_(X) , is given by thefollowing equation (101) based on the above equations (86) and (88). Thelateral force of the tire, F_(Y) , is given by the following equation(102) based on the above equations (87) and (89):

F _(XY)=μ_(max) F _(Z)  (100)

[0100] $\begin{matrix}{F_{X} = {{- \frac{S}{\lambda}}\quad F_{XY}}} & (101) \\{F_{Y} = {{- \frac{K_{\beta}}{K_{S}}}\quad \tan \quad {\beta \cdot \frac{1 + S}{\lambda}}\quad {F_{XY}.}}} & (102)\end{matrix}$

[0101] The following equation (103) is obtained from the above equations(101) and (102), and the following equation (104) is also obtained:$\begin{matrix}{F_{Y} = {\frac{1 + S}{S}\quad \frac{K_{\beta}}{K_{S}}\quad \tan \quad {\beta \cdot F_{X}}}} & (103) \\{\frac{\partial F_{XY}}{\partial\lambda} = 0} & (104)\end{matrix}$

[0102] The above equations (100) to (104) for ξ≦0 respectively representthe values in the case where the tip of the vector of the reaction forceof the road to the tire, F_(XY), is located between the points A and Bor between the points D and E on the critical friction circle 108.

[0103] Note that it can be seen from the above equations (99) and (104)that ∂F_(XY)/∂λ is obtained by obtaining the maximum road frictioncoefficient Ad (see the section 11 below), the vertical load F_(Z) (seethe section 2 above), the slip ratio S (see the section 10 below), thelateral slip angle β of the tire (see the section 8 below), thelongitudinal rigidity K_(S) and lateral rigidity K_(β) of the tire (seethe section 7 below).

[0104] 7. Calculation of the Tire Rigidity

[0105] The longitudinal rigidity K_(S) and lateral rigidity K_(β)of thetire are functions of the reaction force of the road to the tire,F_(XY), and the vertical load F_(Z). It is herein assumed that K_(S) andK_(β)are respectively given by the following equations (105) and (106),provided that K_(XYS) and K_(XYβ)are coefficients of the reaction forceof the road F_(XY), and K_(ZS) and K_(Zβ) are coefficients of thevertical load F_(Z). Note that this assumption does not go against thefacts.

K _(S) =K _(XYS) ·F _(XY) +K _(ZS) ·F _(Z)  (105)

K _(β) =K _(XYβ) ·F _(XY) +K _(Zβ) ·F _(Z)  (106)

[0106] 8. Calculation of the lateral Slip Angle of the Tire of EachWheel

[0107] It is herein assumed that the lateral slip angle of the leftwheel is equal to that of the right wheel. Based on the estimatedvehicle speed VB in the section 10 below, the lateral slip angle β_(B)of the vehicle and the steering angle δ the lateral slip angles β_(fl),β_(fr) of the left and right front wheels (the lateral slip angle of ofthe front wheels) as well as the lateral slip angles β_(rl), β_(rr) ofthe left and right rear wheels (the lateral slip angle γ_(r) of the rearwheels) can be respectively obtained by the following equations (107)and (108) (see FIG. 5): $\begin{matrix}{\beta_{fl} = {\beta_{fr} = {\beta_{f} = {{{\arctan \quad \frac{{{VB}\quad \tan \quad \beta_{B}} + {L_{f}\quad \gamma}}{VB}} - \delta} = {{\arctan \quad \left( {{\tan \quad \beta_{B}} + \frac{L_{f}\quad \gamma}{VB}} \right)} - \delta}}}}} & (107) \\{\beta_{rl} = {\beta_{rr} = {\beta_{r} = {{\arctan \quad \frac{{{VB}\quad \tan \quad \beta_{B}} - {L_{r}\quad \gamma}}{VB}} = {\arctan \quad \left( {{\tan \quad \beta_{B}} - \frac{L_{r}\quad \gamma}{VB}} \right)}}}}} & (108)\end{matrix}$

[0108] Note that the lateral slip angle γ_(B) of the vehicle may becalculated by any method known to those skilled in the art. For example,a deviation of the lateral acceleration as a deviation G_(Y)−Vγof thelateral acceleration G_(Y) from the product Vγ of the vehicle speed Vand the yaw rate γ, that is, lateral slip acceleration V_(Yd) of thevehicle is calculated. The lateral slip velocity V_(Y) of the vehiclemay be calculated by integrating the lateral slip acceleration V_(Yd),and the slip angle γ_(B) of the vehicle may be calculated as a ratio ofthe lateral slip velocity V_(Y) to the longitudinal speed V_(X) of thevehicle (=vehicle speed V), that is, a ratio V_(Y)/V_(x).

[0109] 9. Calculation of the Corrected Wheel Speed The wheel speedVW_(i) of each wheel is converted into the longitudinal speed at thecenter of gravity 104 of the vehicle (hereinafter, referred to as“corrected vehicle speed SVW₁” (i =fl, fr, rl, rr)).

[0110] For example, as shown in FIG. 6, the following equations (109)and (110) are obtained for the left front wheel: $\begin{matrix}{{\frac{{VW}_{fl}}{\cos \quad \beta_{f}}\quad \cos \quad \left( {\delta + \beta_{f}} \right)} = {{SVW}_{fl} - {\frac{Tr}{2}\quad \gamma}}} & (109) \\{{\frac{{VW}_{fl}}{\cos \quad \beta_{f}}\quad \sin \quad \left( {\delta + \beta_{f}} \right)} = {{{SVW}_{fl}\quad \tan \quad \beta_{B}} + {L_{f}{\gamma.}}}} & (110)\end{matrix}$

[0111] Based on the above equations (109) and (110), the corrected wheelspeeds SVW_(fl), SVW_(fr) of the left and right front wheels arerespectively obtained by the following equations (111) and (112):$\begin{matrix}{{SVW}_{fl} = \frac{\begin{matrix}{{{- \left( {{- \frac{Tr}{2}} + {L_{f}\quad \tan \quad \beta_{B}}} \right)}\quad \gamma} +} \\\sqrt{{\left( {1 + {\tan^{2}\quad \beta_{B}}} \right)\quad \left( \frac{{VW}_{fl}}{\cos \quad \beta_{f}} \right)^{2}} - {\left( {L_{f} + {\frac{Tr}{2}\quad \tan \quad \beta_{B}}} \right)^{2}\quad \gamma^{2}}}\end{matrix}}{\left( {1 + {\tan^{2}\quad \beta_{B}}} \right)}} & (111) \\{{SVW}_{fl} = {\frac{\begin{matrix}{{{- \left( {{- \frac{Tr}{2}} + {L_{f}\quad \tan \quad \beta_{B}}} \right)}\quad \gamma} +} \\\sqrt{{\left( {1 + {\tan^{2}\quad \beta_{B}}} \right)\quad \left( \frac{{VW}_{fr}}{\cos \quad \beta_{f}} \right)^{2}} - {\left( {L_{f} - {\frac{Tr}{2}\quad \tan \quad \beta_{B}}} \right)^{2}\quad \gamma^{2}}}\end{matrix}}{1 + {\tan^{2}\quad \beta_{B}}}.}} & (112)\end{matrix}$

[0112] The corrected wheel speeds SVW_(rl), SVW_(rr) of the left andright rear wheels are respectively obtained by the following equations(113) and (114): $\begin{matrix}{{SVW}_{rl} = {{VW}_{rl} + {\frac{Tr}{2}\gamma}}} & (113) \\{{SVW}_{rr} = {{VW}_{rr} - {\frac{Tr}{2}{\gamma.}}}} & (114)\end{matrix}$

[0113] 10. Calculation of the Estimated Vehicle Speed and the Slip Ratioof Each Wheel

[0114] (1) Reference Slip Ratio SK

[0115] The slip ratio for calculating the estimated vehicle speed VB(hereinafter, referred to as “reference slip ratio SK”) is defined asfollows:

[0116] When |F_(X)|is large and |F_(Y)|is large:

[0117] The reference slip ratio SK is given by the following equation(115), based on the above equations (98) and (103) of the tire model:$\begin{matrix}{{SK} = {\frac{\frac{K_{\beta}}{K_{S}}\tan \quad \beta}{\frac{F_{Y}}{F_{X}} - {\frac{K_{\beta}}{K_{S}}\tan \quad \beta}}.}} & (115)\end{matrix}$

[0118] When |F_(X)|is large and |F_(Y)|is small:

[0119] From the above equation (82) of the tire model (where β=0), thereference slip ratio SK is given by the following equation (120) basedon the following equations (116) to (119):

λ=|S|  (116)

[0120] $\begin{matrix}{\xi = {1 - {\frac{K_{S}}{3\mu_{\max}F}{S}}}} & (117)\end{matrix}$

F _(XY)=μ_(max) F _(Z)(1−ξ³)  (118)

|F _(X) |=F _(XY)  (119)

[0121] $\begin{matrix}{{{SK}} = {\left( {1 - {3\sqrt{1 - \frac{F_{X}}{\mu_{\max}F_{Z}}}}} \right){\frac{3\mu_{\max}F_{Z}}{K_{S}}.}}} & (120)\end{matrix}$

[0122] When |F_(X)|is small

[0123] In this case, the reference slip ratio SK is zero. The referenceslip ratio SK is thus given by the following equation (121):

SK=0 . . . (121).

[0124] Accordingly, the reference slip ratio SK (the reference slipratio SK, of each wheel (i=fl, fr, rl, rr)) is calculated bysubstituting for the above equations (115) and (120) the longitudinalforce F_(X) and the like calculated in the sections 2 to 5, 7 and 8above and the section 11 below.

[0125] (2) Estimated Vehicle Speed

[0126] Based on the largest value among the corrected wheel speeds SVW₁,calculated in the section 9 above and the reference slip SK₁ of thatwheel, the estimated vehicle speed VB is calculated according to thefollowing equation (122). The reason why the largest value among thecorrected wheel speeds SVW₁, is used is because this value is theclosest to the actual vehicle speed.

VB =SVW ₁(1+SK ₁)  (122)

[0127] (3) Slip Ratio of Each Wheel

[0128] The slip ratio S₁ of each wheel (i=fl, fr, rl, rr) is calculatedaccording to the following equations (123) to (126), based on theestimated vehicle speed VB and the reference slip ratio SK₁ of eachwheel: $\begin{matrix}{S_{fl} = \frac{{VB} - {SVW}_{fl}}{{SVW}_{fl}}} & (123) \\{S_{fr} = \frac{{VB} - {SVW}_{fr}}{{SVW}_{fr}}} & (124) \\{S_{rl} = \frac{{VB} - {SVW}_{rl}}{{SVW}_{rl}}} & (125) \\{S_{rr} = {\frac{{VB} - {SVW}_{rr}}{{SVW}_{rr}}.}} & (126)\end{matrix}$

[0129] 11. Calculation of the Maximum Road Friction Coefficient of eachWheel

[0130] Based on the vertical load F_(Z) in the section 2 above, thereaction force of the road to the tire F_(XY) in the section 5 above,and the above equations (99) and (104) of the tire model, the maximumroad friction coefficient μ_(max) is given by the following equation(127). Note that, in the equation (127), Δμ is a positive constant, and(∂F_(XY)/∂λ)_(λ=0) is the value (∂F_(XY)/∂λ) when λ=0. $\begin{matrix}\begin{matrix}{\mu_{\max} = {\frac{F_{XY}}{F_{Z}} + {{\Delta\mu}\frac{1}{F_{Z}}\frac{\frac{\partial F_{XY}}{\partial\lambda}}{\frac{1}{F_{Z}}\left( \frac{\partial F_{XY}}{\partial\lambda} \right)_{\lambda = 0}}}}} \\{= {\frac{F_{XY}}{F_{Z}} + {{\Delta\mu}\frac{\frac{\partial F_{XY}}{\partial\lambda}}{\left( \frac{\partial F_{XY}}{\partial\lambda} \right)_{\lambda = 0}}}}}\end{matrix} & (127)\end{matrix}$

[0131] As shown in FIG. 7, (1/F_(Z))(∂F_(XY)/∂λ)_(λ=0) is an inclinationof the μ-λ curve at the origin. (1/F_(Z))(∂F_(XY)/∂λ) is an inclinationof the μ-λ curve for a specific value λ (e.g., λ1). As shown in FIG. 3B,the inclination of the μ-λ curve gradually decreases as the compositeslip ratio λ increases. In the region of the maximum road frictioncoefficient μ_(max), the inclination of the μ-λ curve is zero regardlessof the composite slip ratio λ.

[0132] Accordingly, provided that the minimum value of the compositeslip ratio λ in the region of the maximum road friction coefficient timeis μ_(max) is λe, the ratio between the inclinations of the μ-λ curve inthe second term of the above equation (127) gradually decreases in theregion of λ<λe as the composite slip ratio λ increases. In the regionof, λ≧λe, this ratio is zero. According to the above equation (127), themaximum friction coefficient μ_(max) in the region of λ<λe is estimatedto be a value that is higher than the value F_(XY)/F_(Z) by the value ofthe product of Δμand the aforementioned ratio between the inclinations.In the region of λ≧λe, the maximum friction coefficient μ_(max) isestimated to be a true maximum friction coefficient.

[0133] For example, as shown in FIG. 8, it is herein assumed that thetrue maximum friction coefficient is μ_(true), and the valueF_(XY)/F_(Z) corresponds to the point A1 when the value λ is equal to λ1(λ1≦λe). In this case, the maximum friction coefficient μ_(max) isestimated to be a value corresponding to the point A2, whereby the μ-λcurve is estimated as curve A. In contrast, provided that the valueF_(XY)/F_(Z) corresponds to the point B1 when λ is equal to λ2 (λ2≧λe),the maximum friction coefficient μ_(max) is estimated to be a valuecorresponding to the same point B2 as the point B1, whereby the μ-λcurve is estimated as curve B.

[0134] As can be seen from FIG. 8, the estimation error of the maximumfriction coefficient μ_(max) is increased when λ is small. When theconstant Δμis set to a small value, the maximum friction coefficientμ_(max) is estimated to be a value smaller than the true maximumfriction coefficient μ_(true). In contrast, when the constant Δμis setto a large value, the maximum friction coefficient μ_(max) is estimatedto be a value larger than the true maximum friction coefficientμ_(true). However, the estimation error of the maximum frictioncoefficient gradually decreases as λ increases. In the region of λ≧λe,the maximum friction coefficient μ_(max) is correctly estimated to bethe true maximum friction coefficient μ_(true).

[0135] Note that, according to the equation (83) of the tire model, ξ isequal to 1 when the composite slip ratio λ is zero. In this case, thefollowing equation (128) is obtained: $\begin{matrix}{\left( \frac{\partial F_{XY}}{\partial\lambda} \right)_{\lambda = 0} = {K_{S}.}} & (128)\end{matrix}$

[0136] As described above in the section 6, a current maximum frictioncoefficient μ_(max) is required to calculate ∂F_(XY)/∂λ. Accordingly,afxylax is calculated by using the previous calculated valueμ_(max(n-1)) as the maximum friction coefficient μ_(max) Based on thecalculated value ∂F_(XY)/∂λ, the maximum friction coefficient μ_(max) iscalculated according to the above equation (127).

[0137] Hereinafter, a first embodiment of the invention will bedescribed in detail in conjunction with the accompanying drawings.

[0138]FIG. 9 is a schematic structural diagram showing a maximumfriction coefficient estimating apparatus applied to a rear-wheel-drivevehicle, according to the first embodiment of the invention.

[0139] In FIG. 9, reference numeral 10 denotes an engine. The drivingforce of the engine 10 is transmitted to a propeller shaft 18 through anautomatic transmission 16 that includes a torque converter 12 and atransmission 14. The driving force of the propeller shaft 18 istransmitted to a left rear axle 22L and a right rear axle 22R through adifferential 20. Left and right rear wheels 24RL and 24RR serving asdriving wheels are thus rotated.

[0140] Left and right front wheels 24FL and 24FR serve as driven wheelsas well as steering wheels. Although not shown in FIG. 9, the frontwheels 24FL and 24FR are steered through a tie rod by a rack-and-piniontype power steering device that is driven in response to turning of thesteering wheel by the driver.

[0141] The braking force of the left and right front wheels 24FL, 24FRand the left and right rear wheels 24RL, 24RR is controlled bycontrolling the braking pressure of corresponding wheel cylinders 30FL,30FR, 30RL, 30RR by a hydraulic circuit 28 in a brake system 26.Although not shown in FIG. 9, the hydraulic circuit 28 includes an oilreservoir, an oil pump, various valve devices and the like. The brakingpressure of each wheel cylinder is normally controlled by an electroniccontrol unit (ECU) 36 according to the pressure in a mask cylinder 34that is driven in response to a depressing operation of a brake pedal 32by the driver. The control pressure of each wheel cylinder is controlledby the ECU 36 so as to stabilize the vehicle's behavior as required.

[0142] The ECU 36 receives the following signals: a signal indicatinglongitudinal acceleration G_(X), of the vehicle detected by alongitudinal acceleration sensor 38; a signal indicating lateralacceleration G_(Y) of the vehicle detected by a lateral accelerationsensor 40; a signal indicating a yaw rate γof the vehicle detected by ayaw rate sensor 42; a signal indicating a steering angle δdetected by asteering angle sensor 44; a signal indicating a pressure P₁ (i=fl, fr,rl, rr) in the wheel cylinders 30FL to 30RR of the left and right frontwheels and the left and right rear wheels detected by pressure sensors46FL to 46RR; and a signal indicating a wheel speed VW₁ (i=fl, fr, rl,rr) of the left and right front wheels and the left and right rearwheels detected by vehicle speed sensors 48FL to 48RR.

[0143] Note that the ECU 36 actually includes a CPU (central processingunit), a ROM (read only memory), a RAM (random access memory), and aninput/output (I/O) port device. The ECU 36 may be formed from amicrocomputer of a well-known structure having these elements connectedtogether through a bi-directional common bus, and a driving circuit.

[0144] The ECU 36 stores the control flows of FIGS. 10 and 11. The ECU36 calculates the following values: the longitudinal force F_(X1) andlateral force F_(Y1) of the tire of each wheel described below (i=fl,fr, rl, rr); the reaction force F_(XY1) of the road to each wheel (i=fl,fr, rl, rr) based on the longitudinal and lateral forces F_(X1) andF_(Y1) of the tire; the vertical load F_(Z1) of each wheel (i=fl, fr,rl, rr); and the ratio of variation in reaction force F_(XY) tovariation in composite slip ratio λ for each wheel, ∂F_(XY)/∂μ. Based onthese values, the ECU 36 calculates the maximum friction coefficientμ_(max) for each wheel.

[0145] Although not shown in the figure, the ECU 36 calculates thevalues such as yaw moment M₁ around the center of gravity of the vehicleresulting from the reaction force F_(XY1), of the road (i=fl, fr, rl,rr), and determines the vehicle's behavior based on the calculated yawmoment M₁ and the like. When the vehicle is in a spin state or in adrift-out state, the ECU 36 controls the braking pressure on apredetermined wheel to apply required braking force to the predeterminedwheel, thereby stabilizing the vehicle's behavior. Note that, sincecontrol of the vehicle's behavior based on the yaw moment M₁ and thelike does not form a subject matter of the invention, detaileddescription thereof is omitted. Hereinafter, a routine for calculatingthe maximum friction coefficient according to the first embodiment willbe described with reference to the flowcharts of FIGS. 10 and 11. Notethat control according to the flowcharts of FIGS. 10 and 11 is startedin response to closing of an ignition switch, not shown, and is repeatedat predetermined intervals of time.

[0146] First, in Step S10, signals such as a signal indicating thelongitudinal acceleration G_(X) of the vehicle detected by thelongitudinal acceleration sensor 38 are read. In Step S20, the brakingforce B₁ of each wheel is calculated according to the above equations(4) to (7), based on the braking pressure P₁.

[0147] In Step S30, the wheel acceleration VWd₁, is calculated as a timederivative value of the wheel speed VW₁, and the longitudinal tire forceF_(X1) of each wheel is calculated according to the above equations (17)to (20), based on the wheel acceleration VWd₁, and the like. In StepS40, the driving force D of the vehicle is calculated according to theabove equation (21).

[0148] In Step S50, whether the vehicle is turning to the left or not isdetermined based on, e.g., the sign of the yaw rate γof the vehicledetected by the yaw rate sensor 42. If NO in Step S50, the routineproceeds to Step S80. If YES in Step S50, the routine proceeds to StepS60. Note that determination of the turning state of the vehicle may beconducted by any method that is known in the art.

[0149] In Step S60, the lateral forces of the tires of the left andright front wheels, F_(Yfl) and F_(Yfr), are calculated according to theabove equations (45) and (46) or equations (48) and (49), respectively.In Step S70, the lateral forces of the tires of the left and right rearwheels, F_(Yrl) and F_(Yrr), are calculated according to the aboveequations (68) and (69) or equations (71) and (72), respectively.

[0150] Similarly, in Step S80, the lateral forces of the tire of theleft and right front wheels, F_(Yfl) and F_(Yfr), are calculatedaccording to the above equations (50) and (51) or equations (52) and(53), respectively. In Step S90, the lateral forces of the tires of theleft and right rear wheels, F_(Yrl) and F_(Yrr), are calculatedaccording to the above equations (73) and (74) or equations (75) and(76), respectively.

[0151] In Step S100, the reaction force of the road to each wheel,F_(XY) is calculated according to the above equations (77) to (80),based on the longitudinal and lateral forces F_(X1) and F_(Y1) of thetire of each wheel. In Step S110,the vertical load of each wheel,F_(Z1), is calculated according to the above equations (8) and (11),based on the longitudinal acceleration G_(X) of the vehicle and thelike.

[0152] In Step S120, the ratio of variation in reaction force of theroad F_(XY) to variation in composite slip ratio λ, that is, the ratio∂F_(XY)/∂λis calculated for each wheel according to the routine of FIG.11. In Step S150, the maximum road friction coefficient μ_(max) iscalculated for each wheel according to the above equation (127). Theroutine then returns to Step S10.

[0153] In Step S122 of the routine for calculating the ratio∂F_(XY)/∂λin Step S120 of FIG. 10, the longitudinal rigidity K_(S) andlateral tire rigidity K_(p) of the tire is calculated for each wheelaccording to the above equations (105) and (106). In Step S124, thelateral slip angle μ_(B) of the vehicle is calculated by a method knownin the art, and based on the calculated lateral slip angle μ_(B) thelateral slip angle β₁ of each wheel is calculated according to the aboveequations (107) and (108).

[0154] In Step S126, the corrected vehicle speed SVW₁, of each wheel iscalculated according to the above equations (111) to (114). In StepS128, the reference slip ratio SK₁ of each wheel is calculated accordingto the above equation (115) or (120). In Step S130, the estimatedvehicle speed VB is calculated according to the above equation (122),based on the largest value among the corrected wheel speeds SVW₁.

[0155] In Step S132, the slip ratio S₁, of each wheel is calculatedaccording to the above equations (123) to (126), based on the estimatedvehicle speed VB and the reference slip ratio SK₁ of each wheel. In StepS134, the composite slip ratio λ is calculated according to the aboveequation (82).

[0156] In Step S136, whether the composite slip ratio λ is equal to zeroor not is determined. If NO in Step S136, the routine proceeds to StepS140. If YES in Step S136, the routine proceeds to Step S138. In StepS138, the ratio ∂F_(XY)/∂λ (the ratio of variation in reaction force ofthe road, F_(XY), to variation in composite slip ratio λ) for λ=0, thatis, the ratio (∂F_(XY)/∂λ)₌₀, is set to the vertical rigidity K_(S) ofthe tire. The routine then proceeds to Step S140.

[0157] In Step S140, the value ξis calculated according to the aboveequation (83), based on the previous calculated value of the maximumfriction coefficient μ_(max) and the like. In Step S142, whether thevalue ξis positive or not is determined. If NO in Step S142, the routineproceeds to Step S144, where the ratio ∂F_(XY)/∂λ (the ratio ofvariation in reaction force of the road, F_(XY), to variation incomposite slip ratio λ) is set to zero. If YES in Step S142, the routineproceeds to Step S146, where the ratio ∂F_(XY)/∂λ is calculatedaccording to the above equation (99), based on the previous calculatedvalue of the maximum friction coefficient μ_(max) and the like.

[0158] Although not specifically shown in FIG. 11, Steps S134 to S146are sequentially conducted on a wheel-by-wheel basis in the order of,e.g., the left front wheel, right front wheel, left rear wheel and rightrear wheel. Accordingly, the composite slip ratio λ and the like arecalculated for each wheel.

[0159] According to the first embodiment, the braking force B₁ of eachwheel is calculated in Step S20. The longitudinal force F_(X1) of thetire of each wheel is calculated in Step S30. The driving force D of thevehicle is calculated in Step S40. The lateral force F_(Y1) of the tireof each wheel is calculated in Steps S50 to S90. The reaction force ofthe road to each wheel, F_(XY1), is calculated in Step S100. Thevertical load F_(Z1) of each wheel is calculated in Step S110.

[0160] In Step S120, the ratio ∂F_(XY)/∂λ (the ratio of variation inreaction force of the road, F_(XY), to variation in composite slip ratioλ) is calculated for each wheel. In Step S150, the maximum road frictioncoefficient μ_(max1) is calculated for each wheel according to theequation (127) as the sum of the ratio F_(XY1)/F_(Z1) (the ratio of thereaction force of the road, F_(XY1), to the vertical load F_(Z1)) andthe product of a predetermined coefficient and the ratio ∂F_(XY)/∂λ.

[0161] According to the first embodiment, as described above in thesection 11, an estimated maximum friction coefficient gradually getscloser to the actual maximum friction coefficient as the composite slipratio increases. Accordingly, in the region of the high composite slipratio, the maximum road friction coefficient μ_(max1) can be accuratelyestimated on a wheel-by-wheel basis.

[0162] Note that, in the region of the low composite slip ratio, themaximum road friction coefficient cannot be estimated accurately.However, information on the maximum road friction coefficient isgenerally required when behavior control for stabilizing deterioratedbehavior of the vehicle is to be conducted. The composite slip ratio ishigh in such a situation. Therefore, according to the first embodiment,the maximum road friction coefficient can be accurately estimated in thesituation where the information on the maximum road friction coefficientis required. This enables accurate behavior control. Moreover, such lowestimation accuracy of the maximum road friction coefficient in theregion of the low composite slip ratio will not cause any excessiveinconveniences.

[0163] According to the first embodiment, estimation can be conductedeven when the wheels are not in a predetermined acceleration slip state.Therefore, the maximum road friction coefficient can be accuratelyestimated much more frequently than in the case of the aforementionedconventional estimating apparatus. The maximum road friction coefficientcan also be estimated accurately for the driven wheels.

Second Embodiment

[0164]FIG. 12 is a schematic structural diagram showing a maximumfriction coefficient estimating apparatus applied to a front-wheel-drivevehicle, according to the second embodiment of the invention. FIG. 13 isa flowchart corresponding to FIG. 10, illustrating a routine forestimating the maximum friction coefficient according to the secondembodiment. Note that the same members are denoted with the samereference numerals and characters in FIGS. 9 and 12, and thecorresponding steps are denoted with the same step numbers in FIGS. 10and 13.

[0165] In the second embodiment, the driving force of the engine 10 istransmitted to a left front axle 56L and a right front axle 56R throughthe automatic transmission 16 and a differential 54. Thus, the left andright front wheels 24FL and 24FR serving as steering wheels as well asdriving wheels are rotated.

[0166] In the second embodiment, the longitudinal force F_(X1) of thetire of each wheel is calculated in Step S30 according to the aboveequations (26) to (29). In Step S40, the driving force D of the vehicleis calculated according to the above equation (30). In other respects,the maximum road friction coefficient μ_(max) of each wheel iscalculated in the same manner as that of the first embodiment.

[0167] According to the second embodiment, the maximum road frictioncoefficient μ_(maxi) can be accurately estimated on a wheel-by-wheelbasis in the region of the high composite slip ratio even when thevehicle is a front-wheel-drive vehicle. Moreover, as in the firstembodiment, the maximum road friction coefficient can be accuratelyestimated much more frequently than in the case of the aforementionedconventional estimating apparatus. Accordingly, the maximum roadfriction coefficient can also be estimated accurately for the drivenwheels.

[0168] In particular, according to the illustrated embodiments, acoefficient Δμ·{(∂F_(XY)/∂λ)}_(λ=0) for the ratio ∂F_(XY)/∂λ (the ratioof variation in reaction force of the road, F_(XY), to variation incomposite slip ratio λ) is inversely proportional to the ratio∂F_(XY)/∂λ for λ=0, that is, (∂F_(XY)/∂λ)_(λ=0). Accordingly, themaximum road friction coefficient μ_(max) can be more accuratelyestimated for each wheel as compared to the case where this coefficientis constant.

[0169] The specific embodiments of the invention have been described indetail. However, it should be appreciated by those skilled in the artthat the invention is not limited to the above embodiments, and variousother embodiments are possible without departing from the scope of theinvention.

[0170] For example, the invention is applied to a rear-wheel-drivevehicle in the first embodiment, and applied to a front-wheel-drivevehicle in the second embodiment. However, the invention may be appliedto a four-wheel drive vehicle. In this case, the longitudinal forces ofthe tires of the left and right front wheels, F_(Xfl) and F_(Xfr), andthe longitudinal forces of the tires of the left and right rear wheels,F_(Xrl), F_(Xrr), are respectively calculated according to the followingequations (129) to (132), based on the front-wheel distribution ratioR_(df) and the rear-wheel distribution ratio R_(df) of the driving forceapplied from a four-wheel drive controller: $\begin{matrix}{F_{Xfl} = {B_{fl} + {\frac{1}{2}{D \cdot R_{df}}} - \frac{I_{wf} \cdot {VWd}_{fl}}{r^{2}}}} & (129) \\{F_{Xfr} = {B_{fr} + {\frac{1}{2}{D \cdot R_{df}}} - \frac{I_{wf} \cdot {VWd}_{fr}}{r^{2}}}} & (130) \\{F_{Xrl} = {B_{rl} + {\frac{1}{2}{D \cdot R_{df}}} - \frac{I_{wf} \cdot {VWd}_{rl}}{r^{2}}}} & (131) \\{F_{Xrr} = {B_{rr} + {\frac{1}{2}{D \cdot R_{df}}} - {\frac{I_{wf} \cdot {VWd}_{rr}}{r^{2}}.}}} & (132)\end{matrix}$

[0171] In the above embodiments, the coefficient Δμ·{(∂F_(XY)/∂λ)}_(λ=0)for the ratio ∂F_(XY)/∂λ (the ratio of variation in reaction force ofthe road, F_(XY), to variation in composite slip ratio λ) is set as avalue inversely proportional to the ratio ∂F_(XY)/∂λ for λ=0,i.e.,(∂F_(XY)/∂λ)_(λ=0). However, this coefficient may be set to a fixedvalue.

[0172] In the above embodiments, the longitudinal rigidity K_(S) andlateral rigidity K_(β)of the tire are respectively calculated accordingto the above equations (105) and (106). However, these values may becalculated by another method. The longitudinal rigidity K_(S) andlateral rigidity K_(β)of the tire may be set to a constant.

[0173] As is apparent from the foregoing description, according to theinvention, the maximum road friction coefficient can be calculatedregardless of whether the wheel is in a predetermined drive slip stateor not. Moreover, the maximum road friction coefficient can becalculated either for the driving wheels or driven wheels. Furthermore,the maximum road friction coefficient can be accurately calculated inthe region of the high slip ratio.

[0174] In the illustrated embodiments, the controller is implementedwith a general purpose processor. It will be appreciated by thoseskilled in the art that the controller can be implemented using a singlespecial purpose integrated circuit (e.g., ASIC) having a main or centralprocessor section for overall, system-level control, and separatesections dedicated to performing various different specificcomputations, functions and other processes under control of the centralprocessor section. The controller can be a plurality of separatededicated or programmable integrated or other electronic circuits ordevices (e.g., hardwired electronic or logic circuits such as discreteelement circuits, or programmable logic devices such as PLDs, PLAs, PALsor the like). The controller can be suitably programmed for use with ageneral purpose computer, e.g., a microprocessor, microcontroller orother processor device (CPU or MPU), either alone or in conjunction withone or more peripheral (e.g., integrated circuit) data and signalprocessing devices. In general, any device or assembly of devices onwhich a finite state machine capable of implementing the proceduresdescribed herein can be used as the controller. A distributed processingarchitecture can be used for maximum data/signal processing capabilityand speed.

[0175] While the invention has been described with reference to what arepreferred embodiments thereof, it is to be understood that the inventionis not limited to the preferred embodiments or constructions. To thecontrary, the invention is intended to cover various modifications andequivalent arrangements. In addition, while the various elements of thepreferred embodiments are shown in various combinations andconfigurations, which are exemplary, other combinations andconfigurations, including more, less or only a single element, are alsowithin the spirit and scope of the invention.

What is claimed is:
 1. A controller for determining a maximum frictioncoefficient between a tire of a wheel and a road, comprising: a firstsection that calculates a reaction force of the road to the tire of thewheel based on a tire model; a second section that calculates a verticalload of the tire of the wheel; a third section that calculates as afirst ratio a ratio of the reaction force of the road to the verticalload; a fourth section that calculates as a second ratio a ratio ofvariation in the reaction force of the road to variation in a slip ratioof the tire, the slip ratio being calculated based on the tire model;and a fifth section that calculates a maximum road friction coefficientbased on a product of a predetermined coefficient and the second ratio,and the first ratio.
 2. The controller according to claim 1, wherein thefifth section calculates the maximum road friction coefficient by addingthe first ratio to the product of the predetermined coefficient and thesecond ratio.
 3. The controller according to claim 1, wherein thereaction force of the road is a reaction force in a two-dimensionalplane on the road.
 4. The controller according to claim 1, wherein theslip ratio is a composite slip ratio in a direction of the reactionforce of the road.
 5. The controller according to claim 1, wherein thefirst section further determines a longitudinal force and a lateralforce of the tire of the wheel, and calculates the reaction force of theroad to the tire of the wheel based on the longitudinal force and thelateral force of the tire.
 6. The controller according to claim 5,wherein a series of calculations of the first to fifth sections isrepeatedly conducted at predetermined intervals of time.
 7. Thecontroller according to claim 6, wherein the first section calculatesthe reaction force of the road to the tire of the wheel by using thelongitudinal force of the tire of the wheel that is calculated based ona longitudinal acceleration of a vehicle, a steering angle, a brakingforce of the wheel, and a previous calculated value of the lateral forceof the tire of the wheel.
 8. The controller according to claim 5,wherein the first section calculates the reaction force of the road tothe tire by using a lateral force of a tire of a front wheel that iscalculated based on a yaw rate of a vehicle, a lateral acceleration ofthe vehicle, and the longitudinal force of the tire of the wheel.
 9. Thecontroller according to claim 5, wherein the first section calculatesthe reaction force of the road to the tire by using a lateral force of atire of a rear wheel that is calculated based on a lateral accelerationof a vehicle, the longitudinal force of the tire of the wheel, and alateral force of a tire of a front wheel.
 10. A method for determining amaximum friction coefficient between a tire of a wheel and a road,comprising the steps of: calculating a reaction force of the road to thetire of the wheel based on a tire model; calculating a vertical load ofthe tire of the wheel; calculating as a first ratio a ratio of thereaction force of the road to the vertical load; calculating as a secondratio a ratio of variation in the reaction force of the road tovariation in a slip ratio of the tire, the slip ratio being calculatedbased on the tire model; and calculating a maximum road frictioncoefficient based on a product of a predetermined coefficient and thesecond ratio, and the first ratio.
 11. The method according to claim 10,wherein the maximum road friction coefficient is calculated by addingthe first ratio to the product of the predetermined coefficient and thesecond ratio.
 12. The method according to claim 10, wherein the reactionforce of the road is a reaction force in a two-dimensional plane on theroad.
 13. The method according to claim 10, wherein the slip ratio is acomposite slip ratio in a direction of the reaction force of the road.14. The method according to claim 10, wherein the step of calculatingthe vertical load of the tire of the wheel includes the steps ofestimating a longitudinal force and a lateral force of the tire of thewheel, and calculating the reaction force of the road to the tire of thewheel based on the longitudinal force and the lateral force of the tire.15. The method according to claim 14, wherein a series of the steps inthe method is repeatedly conducted at predetermined intervals of time.16. The method according to claim 15, wherein the step of calculatingthe vertical load of the tire of the wheel includes the step ofcalculating the reaction force of the road to the tire of the wheel byusing the longitudinal force of the tire of the wheel that is calculatedbased on a longitudinal acceleration of a vehicle, a steering angle, abraking force of the wheel, and a previous calculated value of thelateral force of the tire of the wheel.
 17. The method according toclaim 14, wherein the step of calculating the vertical load of the tireof the wheel includes the step of calculating the reaction force of theroad to the tire by using a lateral force of a tire of a front wheelthat is calculated based on a yaw rate of a vehicle, a lateralacceleration of the vehicle, and the longitudinal force of the tire ofthe wheel.
 18. The method according to claim 14, wherein the step ofcalculating the vertical load of the tire of the wheel includes the stepof calculating the reaction force of the road to the tire by using alateral force of a tire of a rear wheel that is calculated based on alateral acceleration of a vehicle, the longitudinal force of the tire ofthe wheel, and a lateral force of a tire of a front wheel.
 19. Ancontroller for determining a maximum friction coefficient between a tireof a wheel and a road, comprising: a first section that calculates areaction force of the road to the tire; a second section that calculatesa vertical load of the tire; a third section that calculates variationin a slip ratio of the tire; a fourth section that calculates variationin the reaction force of the road to the tire; and a fifth section thatcalculates a maximum road friction coefficient based on the reactionforce of the road, the vertical load, the variation in the slip ratio,and the variation in the reaction force of the road.